How do I get specular and bump maps or normal maps to work with a texture that has a transparent background?
Your next step is to make something called an transmission map. This is a black-and-white image that tells Blender which parts you want to let light through, and how much of it. In your case, your hair will be black and the transparent part will be white. 

While I don't know your node setup (it would be helpful to see it to simplify your task), the next few steps should still be applicable. Add a Mix Shader and plug your current setup to the top input. Add a Transparent Shader and add it to the bottom input of the Mix Shader. 

Running under the assumption you already have a bump map available, you'll need to add a total of three more nodes before you can use it. While you probably already have a fair idea of many, if not all of the following instructions (given you're already using a texture map), I'll go through them all, anyway, in case something got left out. The first node you want is a Texture Coordinate node. You should find this under the Input of your Add menu. You'll need this so Blender knows where you want the image to go on your hair. The next one is an Image Texture. You can find it under Texture on the Add menu. This is for actually bringing your map into Blender. Change the first dropdown setting from 'Color' to 'Non-Color Data' and connect a line from the UV output from the Texture Coordinate node into the Vector input on the Image Texture node. Now upload your texture by clicking 'open' and browsing for wherever you saved your map. The final node you'll need is a Bump Map. You'll see this under the Vector section of your Add menu. If your map is black-and-white, hook it into the Height input of the Bump Map. If it's colorful, it's actually a normal map, and you need to hook it into normal input. Finally, hook your Bump Map output to the Vector input of whatever node you have your hair hooked up to.

I can't really help much further until I can see your setup, but I hope this helps so far.